Algebraic and Combinatorial Bounds for the Hat Guessing Number of Graphs

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Shariati, Mohammad Reza | 2023

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 56798 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ebrahimi Boroojeni, Javad
Abstract:
A number of players are positioned on the vertices of a finite graph, and each is wearing a hat whose color has been randomly and equally likely selected from a finite set of colors. Player A sees Player B if and only if there is a directed edge from the vertex corresponding to A to the vertex corresponding to B. The players must agree on a specific strategy before the game begins and carry it out. Once the game begins, no communication is allowed between the players, and each player must simultaneously guess the color of their own hat. The team wins if at least one guess is correct. For a given directed graph, we define the 'guessing number' of the graph as the maximum number of colors for which a winning strategy exists, regardless of the arrangement of the colors. In this study, we use combinatorial, probabilistic, and algebraic techniques to calculate the guessing number for a diverse family of graphs, and we find bounds for the guessing number and the linear guessing number based on the properties of the graph
Keywords:
Lovasz Local Lemma ; Combinatorial Nullstellensatz ; Hat Guessing Strategies ; Hat Guessing Problem ; Hat Guessing Game ; Hat Guessing Graph Number